Recall the equivalence \(D_{\text{sg}}(\mathcal {X}(d)_0)\cong \text{MF}(\mathcal {X}(d), W)\)  and the exact sequence (REF ).
By [1]}, the pushforward \(i_*:\mathcal {X}(d)_0\rightarrow \mathcal {X}(d)\)  induces an algebra morphism
\(i_*: \text{KHA}(Q,W)\rightarrow \text{KHA}(Q,0).\) 
